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- 저자 : Nedela, Roman (검색결과 4 건)
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Classification of regular embeddings of hypercubes of odd dimension
Du, Shao-Fei,Kwak, Jin Ho,Nedela, Roman Elsevier 2007 Discrete mathematics Vol.307 No.1
<P><B>Abstract</B></P><P>By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the <I>n</I>-dimensional cubes <SUB>Qn</SUB> into nonorientable surfaces exist for any positive integer n>2. In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807–823] presented a construction giving for each solution of the congruence <SUP>e2</SUP>≡1(modn) a regular embedding <SUB>Me</SUB> of the hypercube <SUB>Qn</SUB> into an orientable surface. It was conjectured that all regular embeddings of <SUB>Qn</SUB> into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes <SUB>Qn</SUB> into orientable surfaces for <I>n</I> odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd <I>n</I>.</P>
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REGULAR MAPS-COMBINATORIAL OBJECTS RELATING DIFFERENT FIELDS OF MATHEMATICS
Nedela, Roman Korean Mathematical Society 2001 대한수학회지 Vol.38 No.5
Regular maps and hypermaps are cellular decompositions of closed surfaces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familar examples of regular maps. The gret dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [59] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [19], and more recently in Coxeter’s and Moser’s book [25] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck\`s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas, Many of these links are nicely surveyed in the recent papers of Jones [55] and Jones and Singerman [54]. The presented survey paper is based on the talk given by the author at the conference “Mathematics in the New Millenium”held in Seoul, October 2000. The idea was, on one hand side, to show the relationship of (regular) maps and hypermaps to the above mentioned fields of mathematics. On the other hand, we wanted to stress some ideas and results that are important for understanding of the nature of these interesting mathematical objects.
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Regular embeddings of complete multipartite graphs
Du, Shao-Fei,Kwak, Jin Ho,Nedela, Roman Elsevier 2005 European journal of combinatorics : Journal europ& Vol.26 No.3
<P><B>Abstract</B></P><P>In this paper, we classify all regular embeddings of the complete multipartite graphs <I>K</I><SUB><I>p</I>,…,<I>p</I></SUB> for a prime <I>p</I> into orientable surfaces. Also, the same work is done for the regular embeddings of the lexicographical product of any connected arc-transitive graph of prime order <I>q</I> with the complement of the complete graph of prime order <I>p</I>, where <I>q</I> and <I>p</I> are not necessarily distinct. Lots of regular maps found in this paper are Cayley maps.</P>
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